Quantum Monte Carlo calculations of the surface energy of an electron gas

We present quantum Monte Carlo calculations of the surface energy of the electron gas (jellium). Our results agree with the best estimates obtained by other methods, thus appearing to resolve the controversy which currently exists and paving the way for future simulations of real surface systems.

Figure 1

(Color online) The components of the jellium surface energy for (a) $r_s=2.07$ and (b) $r_s=3.94$, calculated from Eq. (1) using the LDA. The total surface energy is ${\sigma }_{\mathrm{tot}}$, which is the sum of kinetic $\left({\sigma }_S\right)$, electrostatic $\left({\sigma }_{\mathrm{el}}\right)$, and exchange-correlation $\left({\sigma }_{\mathrm{XC}}\right)$ terms. Note the cancellation between different components of the total energy, especially for $r_s=2.07$.

Figure 2

(Color online) The total energy per electron ${\epsilon }_{\text{slab}}$, calculated for a finite cell using the LDA with (a) $r_s=2.07$ and (b) $r_s=3.94$. $N$ is the number of electrons. For a given density, the abscissa is inversely proportional to the in-plane area of the cell because $s∕N=4\pi r_s^3∕3L^2$. The horizontal lines are the infinite-$L$ limits. The slab widths are chosen so that the infinite-cell surface energy (as calculated in the LDA) matches the infinite-cell, infinite-$s$ limit. The finite-cell errors are more severe when $r_s=2.07$, which means that we are forced to use larger cells at this density.

Figure 3

(Color online) The energy per electron in QMC, as a function of the in-plane cell size, for (a) $r_s=2.07$ and (b) $r_s=3.94$.

Figure 4

(Color online) The difference $\Delta {\epsilon }_{\text{slab}}$ in the energy per electron in slabs of different widths, for (a) $r_s=2.07$ and (b) $r_s=3.94$. The QMC results (extrapolated to the infinite-$L$ limit) are compared with those from the RPA and RPA+.

Figure 5

(Color online) The surface energy of jellium as a function of density. The points correspond approximately to the electron densities of a range of metals: $r_s=2.07$ (Al), 2.30 (Zn), 2.66 (Mg), 3.25 (Li), and 3.94 (Na).